pith. sign in

arxiv: 1906.12167 · v1 · pith:NKPEFPPFnew · submitted 2019-06-26 · 💻 cs.CV

Gray Level Image Threshold Using Neutrosophic Shannon Entropy

Vasile Patrascu This is my paper

Pith reviewed 2026-05-25 16:16 UTC · model grok-4.3

classification 💻 cs.CV
keywords image segmentationgray level thresholdingneutrosophic entropyShannon entropymultilevel thresholdingimage processingneutrosophic sets
0
0 comments X

The pith

Minimizing neutrosophic Shannon entropy yields optimal gray level thresholds for image segmentation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a segmentation method that selects gray level thresholds by minimizing a version of Shannon entropy expressed in neutrosophic terms. The truth, neutrality, and falsity components are defined from how pixels belong to the resulting regions and to the zone around each candidate threshold. This construction supports single or multiple thresholds and is demonstrated on sample grayscale images. A reader would care if the neutrosophic formulation produces thresholds that handle boundary uncertainty more stably than ordinary entropy minimization.

Core claim

The method defines the neutrosophic truth, neutrality, and falsity degrees based on pixel belonging to the segmented regions and the separation threshold area, then minimizes the corresponding Shannon entropy to determine the optimal gray level thresholds for image segmentation.

What carries the argument

Neutrosophic Shannon entropy whose truth, neutrality, and falsity components are defined from region membership and threshold area.

If this is right

  • The same minimization can return several thresholds when the image contains more than two intensity classes.
  • Threshold selection depends only on the intensity histogram and the neutrosophic membership rules.
  • The resulting partitions are intended to respect both region coherence and the uncertain zone near each threshold.
  • Performance is reported as good on the test images used in the experiments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same membership rules could be applied to other entropy measures or to color channels independently.
  • If the neutrality component captures boundary pixels effectively, the method might reduce sensitivity to small intensity shifts compared with crisp entropy.
  • The construction offers a concrete way to blend classical information-theoretic thresholding with neutrosophic uncertainty handling.

Load-bearing premise

The chosen definitions of the neutrosophic components produce an entropy minimum that corresponds to segmentation thresholds that are meaningfully better or more stable than those from standard methods.

What would settle it

Apply the procedure to standard test images, compute segmentation quality metrics such as boundary error or region overlap against ground truth, and compare the scores directly to those obtained by ordinary Shannon entropy thresholding; if the neutrosophic version is not at least as good, the performance claim does not hold.

Figures

Figures reproduced from arXiv: 1906.12167 by Vasile Patrascu.

Figure 1
Figure 1. Figure 1: The graphic of the functions T(red), I(blue), F(green) for v1 = 0.15, v2 = 0.75 and t = 0.3. 3 The Shannon entropy for neutrosophic infor￾mation In this paper, the Shannon function [10] is used as a measure for the neutro￾sophic information uncertainty. We do the following notations: The bifuzzy undefinedness U: U(x, t) = max(0, 1 − T(x, t) − F(x, t)) (13) The bifuzzy contradiction C: C(x, t) = max(0, T(x,… view at source ↗
Figure 2
Figure 2. Figure 2: The image ball (a). The segmented image with two gray levels (b) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The graphic of neutrosophic Shannon entropy where the red circle [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The image block (a). The segmented image with three gray levels (b) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The graphic of neutrosophic Shannon entropy where the red circles [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The image mammography (a). The segmented image with three gray levels (b) [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The graphic of neutrosophic Shannon entropy where the red circles [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The image spider (a). The segmented image with three gray levels (b) [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The graphic of neutrosophic Shannon entropy where the red circles [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
read the original abstract

This article presents a new method of segmenting grayscale images by minimizing Shannon's neutrosophic entropy. For the proposed segmentation method, the neutrosophic information components, i.e., the degree of truth, the degree of neutrality and the degree of falsity are defined taking into account the belonging to the segmented regions and at the same time to the separation threshold area. The principle of the method is simple and easy to understand and can lead to multiple thresholds. The efficacy of the method is illustrated using some test gray level images. The experimental results show that the proposed method has good performance for segmentation with optimal gray level thresholds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes a new grayscale image segmentation method that minimizes Shannon entropy in a neutrosophic framework. The truth, indeterminacy, and falsity components are defined by considering pixel membership in the segmented regions together with the separation threshold area; the approach is presented as simple, able to yield multiple thresholds, and effective on the basis of qualitative illustrations on a small set of test images.

Significance. A validated neutrosophic-entropy criterion could supply an alternative thresholding objective that explicitly models uncertainty, potentially useful in applications where classical entropy or variance-based methods are sensitive to noise or ambiguous boundaries. The current manuscript, however, supplies neither the explicit component formulas, the optimization procedure, quantitative performance metrics, nor any baseline comparison, so the practical significance cannot yet be assessed.

major comments (2)
  1. [Abstract] Abstract: the claim that 'the experimental results show that the proposed method has good performance for segmentation with optimal gray level thresholds' is unsupported; no numerical metrics (e.g., Dice, PSNR, misclassification error), no statistical tests, and no comparison against Otsu, Kapur, or standard Shannon entropy appear in the text.
  2. [Abstract] Abstract: the neutrosophic components are stated to be 'defined taking into account the belonging to the segmented regions and at the same time to the separation threshold area,' yet the manuscript provides neither the explicit mapping from pixels to T/I/F values nor the resulting entropy expression; without these it is impossible to determine whether the minimization is independent of the modeling choices or partly circular.
minor comments (1)
  1. The manuscript would benefit from a dedicated section or appendix containing the full mathematical definitions, the optimization algorithm (including any search strategy for multiple thresholds), and pseudocode to permit reproduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We agree that the current version requires substantial improvements in quantitative validation and explicit mathematical detail to support the claims made. We address each major comment below and will incorporate the necessary revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that 'the experimental results show that the proposed method has good performance for segmentation with optimal gray level thresholds' is unsupported; no numerical metrics (e.g., Dice, PSNR, misclassification error), no statistical tests, and no comparison against Otsu, Kapur, or standard Shannon entropy appear in the text.

    Authors: We agree that the abstract claim is not supported by quantitative evidence in the current manuscript, which relies solely on qualitative illustrations. In the revised version we will add numerical performance metrics (including misclassification error) together with comparisons against Otsu and Kapur methods, and we will revise or remove the unsupported statement in the abstract accordingly. revision: yes

  2. Referee: [Abstract] Abstract: the neutrosophic components are stated to be 'defined taking into account the belonging to the segmented regions and at the same time to the separation threshold area,' yet the manuscript provides neither the explicit mapping from pixels to T/I/F values nor the resulting entropy expression; without these it is impossible to determine whether the minimization is independent of the modeling choices or partly circular.

    Authors: The manuscript provides only a high-level conceptual description of the T/I/F components without the explicit pixel-to-membership mappings or the derived entropy formula. We will add the complete mathematical definitions, the explicit component formulas, and the entropy expression in the revised manuscript to enable reproducibility and independent assessment. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The method defines neutrosophic T/I/F components as functions of region membership and the candidate threshold, then minimizes the resulting Shannon entropy to select the threshold. This is a standard variational optimization construction (objective depends on the decision variable) and does not reduce to a tautology or self-definition by construction. No load-bearing self-citations, imported uniqueness theorems, or ansatz smuggling appear in the supplied text. The central claim rests on experimental demonstration rather than an internal reduction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of extending Shannon entropy to neutrosophic components whose definitions are tied directly to the threshold variable; no free parameters or new entities are declared in the abstract.

axioms (1)
  • domain assumption Shannon entropy remains a meaningful information measure when its arguments are neutrosophic triples (truth, neutrality, falsity).
    The method presupposes this extension without additional justification in the abstract.

pith-pipeline@v0.9.0 · 5622 in / 1193 out tokens · 64399 ms · 2026-05-25T16:16:22.590355+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The neutrosophic information components... are defined taking into account the belonging to the segmented regions and at the same time to the separation threshold area... E(t) = (eT(t) + eI(t) + eF(t))/3. The segmentation thresholds are the local minimum points of the total entropy E.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    Atanassov

    K. Atanassov. Intuitionistic Fuzzy sets. Fuzzy Sets and Systems 20, pp. 87-96, (1986). MDPI, Preprints 2019, 2019060248. 6 The 3rd Conference on Recent Advances in Artificial Intelligence, RAAI 2019 Bucharest, Romˆ ania, June 28-30, 2019. a) b) Figure 6: The image mammography ( a). The segmented image with three gray levels (b). Figure 7: The graphic of ne...

    work page 1986

  2. [2]

    Ashbacher, Introduction to Neutrosophic Logic, American Research Press, Rehoboth, NM, (2002)

    C. Ashbacher, Introduction to Neutrosophic Logic, American Research Press, Rehoboth, NM, (2002)

    work page 2002

  3. [3]

    V. Patrascu, Shannon entropy for imprecise and under-defined or over- defined information, The 25th Conference on Applied and Industrial Math- ematics, CAIM 2017, Iasi, Romania, ROMAI Journal, vol. 14, no. 1, pp. 169-185, (2018)

    work page 2017

  4. [4]

    Patrascu, Shannon Entropy for Neutrosophic Information, doi:10.13140 /RG.2.2.32352.74244, arXiv:1810.00748, September, (2018)

    V. Patrascu, Shannon Entropy for Neutrosophic Information, doi:10.13140 /RG.2.2.32352.74244, arXiv:1810.00748, September, (2018)

    work page arXiv 2018

  5. [5]

    V. Patrascu, A Novel Penta-Valued Descriptor for Color Clustering, The 6th International Conference on Image and Signal Processing, ICISP 2014, Cherbourg, Normandy, France, June 30 - July 2, 2014, Volume: Image and Signal Processing, Lecture Notes in Computer Science, Volume 8509, Springer International Publishing Switzerland, pp. 173-182, (2014). MDPI, P...

    work page 2014

  6. [6]

    V. Patrascu, New Framework of HSL System Based Color Clustering Al- gorithm, The 24th Midwest Artificial Intelligence and Cognitive Sciences Conference, MAICS 2013, April 13-14, 2013, New Albany, Indiana. USA, ceur-ws.org, Vol. 1348, pp. 85-91, (2013)

    work page 2013

  7. [7]

    Rivieccio, Neutrosophic Logics: Prospects and Problems, Fuzzy Sets and Systems, 159, pp

    U. Rivieccio, Neutrosophic Logics: Prospects and Problems, Fuzzy Sets and Systems, 159, pp. 1860-1868, (2008)

    work page 2008

  8. [8]

    Smarandache, A Unifying Field in Logics: Neutrosophic Logic

    F. Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neu- trosophy, Neutrosophic Set, Neutrosophic Probability, American Research Press, Rehoboth, NM, (1999)

    work page 1999

  9. [9]

    Smarandache, Neutrosophic Set - A Generalization of the Intuitionistic Fuzzy Set, International Journal of Pure and Applied Mathematics, 24, no

    F. Smarandache, Neutrosophic Set - A Generalization of the Intuitionistic Fuzzy Set, International Journal of Pure and Applied Mathematics, 24, no. 3, pp. 287-297, (2005)

    work page 2005

  10. [10]

    Shannon, A mathematical theory of communication, Bell System Tech., J

    C.E. Shannon, A mathematical theory of communication, Bell System Tech., J. 27, pp. 379-423, (1948). MDPI, Preprints 2019, 2019060248. 8 The 3rd Conference on Recent Advances in Artificial Intelligence, RAAI 2019 Bucharest, Romˆ ania, June 28-30, 2019

    work page 1948

  11. [11]

    Wikipedia, https://en.wikipedia.org/wiki/Multiset

    work page

  12. [12]

    A. L. Zadeh. Fuzzy sets, Information and Control, 8, pp. 338-353, (1965). MDPI, Preprints 2019, 2019060248. 9

    work page 1965